Hiroyasu Izeki, Shin Nayatani
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the action to have a global fixed point.
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We prove that a harmonic quasi-isometric map between pinched Hadamard surfaces is a quasi-conformal diffeomorphism.
A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of hypergeometric type is obtained with respect to volume density.
We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms
In a previous paper, we proved that a quasi-isometric map $f:X\longrightarrow Y$ between two pinched Hadamard manifolds $X$ and $Y$ is within bounded distance from a unique harmonic map. We extend this result to maps $f:Γ\backslash X\longrightarrow Y$, where $Γ$ is a convex cocompact discrete group of isometries of $X$ and $f$ is locally quasi-isometric at infinity.
In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.
We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds.
We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times R$ over domains of $M$ bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in $M\times R$ with the conformal structure of $C$.
We show that every one-harmonic map, in the sense of Trapani and Valli, from a Hadamard surface of pinched negative curvature to $\mathbb{H}^2$ has image the interior of the convex hull of a subset of $\partial_\infty\mathbb{H}^2$. The proof relies on Minkowski geometry, by interpreting one-harmonic maps as the Gauss maps of convex surfaces.
Harmonic maps to Hadamard spaces and a universal higher Teichmüller space
提出稳定性准则，使Ricci曲率下有界的完备流形到Hadamard空间的粗Lipschitz映射在有限距离内为调和映射，并证明唯一性，从而推广Schoen-Li-Wang猜想并定义通用Hitchin分量。
Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds
本文在旋转对称 Cartan-Hadamard 流形上引入全局频率函数，并在实双曲空间中证明相关频率函数的单调性，进而得到对数凸性定理和双曲三球定理。
In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $Ω$ and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ denote a triangulation of $Ω\cup\bordΩ$. We construct a \emph{new} decomposition of $Ω\cup\bordΩ$ into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a {\it pair} of harmonic functions on ${\mathcal T}^{(0)}$ and properties of their level curves. In the sequel \cite{Her3} it will be proved that a particular discrete schem…
We prove that a quasi-isometric map, and more generally a coarse embedding, between pinched Hadamard manifolds is within bounded distance from a unique harmonic map.
Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces
本文通过调和平均曲率流在Cartan-Hadamard 3空间中证明了锐利的Minkowski型不等式，改进了超曲面总平均曲率的已知估计，特别改进了Ghomi-Spruck的结果。
We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, ${\rm CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist-Hulin.
Estimates of the Poisson kernel on negatively curved Hadamard manifolds
本文研究夹挤负曲率 Hadamard 流形的 Poisson 核，借助 Anderson-Schoen 的正调和函数锥估计证明其整体上下界，并用于量化调和测度向边界点测度及无穷远边界测度的收敛。
We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isomet…
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan-Hadamard spaces of strictly negative curvature, by the existen…
We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence and uniqueness properties of the corresponding harmonic maps. We show that some important applications of smooth harmonic maps can be obtained in this setting.
We give a proof, using harmonic maps from disks to real trees, of Skora's theorem (Morgan-Otal (1993), Skora (1990), originally conjectured by Shalen): if G is the fundamental group of a surface of genus at least 2, then any small minimal G-action on a real tree is dual to the lift of a measured foliation. Analytic tools like the maximum principle are used to simplify the usual combinatorial topology arguments. Other analytic objects associated to a harmonic map, such as the Hopf differential and the moduli space of harmonic maps, are also introduced as tools for understanding the action of su…
Remarks on the generalised Calabi-Yau problem in higher codimension
该研究引入更灵活的凸性概念，得到适用于高余维极小子流形全测地投影无界性研究的新型Omori-Yau最大值原理，并将其应用于多类映射与相关定理的探究，推进广义Calabi-Yau问题相关研究。
Liouville theorem for $V$-harmonic maps under non-negative $(m, V)$-Ricci curvature for non-positive $m$
本文证明了在非负$(m, V)$-Ricci曲率条件下多种增长$V$-调和映射的刘维尔定理，推广了Cheng、Hildebrandt-Jost-Wideman和Choi等人的经典结果，并建立了刘维尔性质与$Δ_V$-扩散过程常返性的联系。
Harmonic projections in negative curvature II: large convex sets
本文在尖顶负曲率流形上，通过非坍缩条件证明调和映射的存在性，并推广到足够大的凸集投影情形，涵盖双曲空间中正则边界开集的凸包。
We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $(n-2)$-skeleton, we improve the regularity to locally Lipschitz. Finally, for points $x \in X^{(k)}$ with $k \leq n-2$, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of $x \in X$.
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